The Stacks project

Lemma 61.22.1. Let $k$ be a field. Let $G = \text{Gal}(k^{sep}/k)$ be its absolute Galois group. Further,

  1. let $M$ be a profinite abelian group with a continuous $G$-action, or

  2. let $\Lambda $ be a Noetherian ring and $I \subset \Lambda $ an ideal an let $M$ be an $I$-adically complete $\Lambda $-module with continuous $G$-action.

Then there is a canonical sheaf $\underline{M}^\wedge $ on $\mathop{\mathrm{Spec}}(k)_{pro\text{-}\acute{e}tale}$ associated to $M$ such that

\[ H^ i(\mathop{\mathrm{Spec}}(k), \underline{M}^\wedge ) = H^ i_{cont}(G, M) \]

as abelian groups or $\Lambda $-modules.

Proof. Proof in case (2). Set $M_ n = M/I^ nM$. Then $M = \mathop{\mathrm{lim}}\nolimits M_ n$ as $M$ is assumed $I$-adically complete. Since the action of $G$ is continuous we get continuous actions of $G$ on $M_ n$. By √Čtale Cohomology, Theorem 59.56.3 this action corresponds to a (locally constant) sheaf $\underline{M_ n}$ of $\Lambda /I^ n$-modules on $\mathop{\mathrm{Spec}}(k)_{\acute{e}tale}$. Pull back to $\mathop{\mathrm{Spec}}(k)_{pro\text{-}\acute{e}tale}$ by the comparison morphism $\epsilon $ and take the limit

\[ \underline{M}^\wedge = \mathop{\mathrm{lim}}\nolimits \epsilon ^{-1}\underline{M_ n} \]

to get the sheaf promised in the lemma. Exactly the same argument as given in the introduction of this section gives the comparison with Tate's continuous Galois cohomology. $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 09B4. Beware of the difference between the letter 'O' and the digit '0'.